Problem: Colin is painting figurines. He spends 20 minutes painting each figurine. After painting for 60 minutes, he still has 9 more figurines left to paint. The number $f$ of figurines left to paint is a function of $t$, the amount of time in minutes Colin spends painting. Write the function's formula. $f=$
Answer: The rate at which Colin paints is constant, so we're dealing with a linear relationship. We could write the desired formula in slope-intercept form: $f= mt+ b$. In this form, $ m$ gives us the slope of the graph of the function and $ b$ gives us the $y$ -intercept. Our goal is to find the values of $ m$ and $ b$ and substitute them into this formula. We know that the number of figurines left for Colin to paint decreases by $1$ every $20$ minutes, so the slope $ m$ is ${-\dfrac{1}{20}}$, and our function looks like $f={-\dfrac{1}{20}}t+ b$. We also know that Colin has $9$ figurines left to paint after painting for $60$ minutes, which means that when $t=60$, $f=9$. We can substitute this into the formula of the function to find $ b$ : $\begin{aligned}{-\dfrac{1}{20}}\cdot60+ b&=9\\\\ -3+ b&=9\\\\ b&={12}\end{aligned}$ This means Colin initially had $12$ figurines to paint. Since $ m = {-\dfrac{1}{20}}$ and $ b = {12}$, the desired formula is: $f={-\dfrac{1}{20}}t+{12}$